Poker/Omaha/Probability derivations for making low hands in Omaha hold 'em

The probability derivations for starting hands making a qualifying low hand in Omaha hold 'em Hi-Lo are separate for each of the starting hand low shapes. The derivations require identifying the individual cases that complete a low hand; make a low hand possible on the board, but don't make a low hand; and that create a board with fewer than three low ranks, making no low hand possible.

The ranks on the board are indicated using lower case letters for matches with the starting hand and upper case letters to indicate qualifying low ranks that don't match the starting hand. So the low hand shape xxyz is any hand with a low pair of x with two additional low ranks y and z and the board xyA represents a flop that contains one x, one of the non-paired ranks y and one other low rank A. Note that since y and z have an identical relationship to the starting hand—each represents an unpaired rank—xyA and xzA represent the same set of boards and are interchangeable, so derivations for this hand choose one of the two choices represented by y. In addition to the upper and lower case letters, * is used to represent any rank higher than the qualifying low. So for the low hand shape xxy*, the board xx* represents a flop that contains two xs and a high rank (either the same high rank as xxy* or a different high rank).

One common calculation regardless of the hand is that no low is possible when there are two or fewer low cards on the board. If h{\displaystyle h} is the number of high cards in the hand (h{\displaystyle h} = 0, 1, 2, 3, 4) and r{\displaystyle r} is the low qualifier (r{\displaystyle r} = 8, 9), then 4r−4+h{\displaystyle 4r-4+h} is the number of low cards remaining in the deck and 52−4r−h{\displaystyle 52-4r-h} is the number of high cards remaining in the deck. Using ? to represent any low card, the number of boards with 0, 1 or 2 low cards on the flop, turn and river are calculated as follows:

Each table in the following subsections shows all of the boards that can make each hand and the derivation for the combinations for that board. Probabilities are determined by dividing the number of combinations for each hand by the (483)=17,296{\displaystyle {\begin{matrix}{48 \choose 3}=17,296\end{matrix}}} boards on the flop, (484)=194,580{\displaystyle {\begin{matrix}{48 \choose 4}=194,580\end{matrix}}} boards on the turn, and (485)=1,712,304{\displaystyle {\begin{matrix}{48 \choose 5}=1,712,304\end{matrix}}} boards at the river. The probabilities for the boards in each table total 1.0.

Starting hands with four high cards (****) cannot make a low hand, so the only possibilities are having a low hand possible on the board or not. With four high cards in the hand, there are 52−4r−4=48−4r{\displaystyle 52-4r-4=48-4r} high cards and 4r−4+4=4r{\displaystyle 4r-4+4=4r} low cards left in the deck. The following tables show the derivations when holding four high cards.

Derivations for making low hands with one low card and three high cards[edit]

Starting hands with one low card and three high cards (x***) cannot make a low hand, so the only possibilities are having a low hand possible on the board or not. With three high cards in the hand, there are 52−4r−3=49−4r{\displaystyle 52-4r-3=49-4r} high cards and 4r−4+3=4r−1{\displaystyle 4r-4+3=4r-1} low cards left in the deck. The following tables show the derivations when holding one low card and three high cards.

Derivations for low hand shape x*** (one low card and three high cards) on the flop

Derivations for making low hands with a low pair and two high cards[edit]

Starting hands with a low pair and two high cards (xx**) cannot make a low hand, so the only possibilities are having a low hand possible on the board or not. With two high cards in the hand, there are 52−4r−2=50−4r{\displaystyle 52-4r-2=50-4r} high cards and 4r−4+2=4r−2{\displaystyle 4r-4+2=4r-2} low cards left in the deck. The following tables show the derivations when holding a low pair and two high cards.

Derivations for low hand shape xx** (a low pair and two high cards) on the flop

Derivations for making low hands with a low three of a kind and a high card[edit]

Starting hands with a low three of a kind and a high card (xxx*) cannot make a low hand, so the only possibilities are having a low hand possible on the board or not. With one high card in the hand, there are 52−4r−1=51−4r{\displaystyle 52-4r-1=51-4r} high cards and 4r−4+1=4r−3{\displaystyle 4r-4+1=4r-3} low cards left in the deck. The following tables show the derivations when holding a low three of a kind and a high card.

Derivations for low hand shape xxx* (a low three of a kind and a high card) on the flop

Starting hands with a low four of a kind (xxxx) cannot make a low hand, so the only possibilities are having a low hand possible on the board or not. With no high cards in the hand, there are 52−4r−0=52−4r{\displaystyle 52-4r-0=52-4r} high cards and 4r−4+0=4r−4{\displaystyle 4r-4+0=4r-4} low cards left in the deck. The following tables show the derivations when holding a low four of a kind.

Derivations for low hand shape xxxx (a low four of a kind) on the flop

Derivations for making low hands with two low ranks and two high cards[edit]

Starting hands with two low ranks and two high cards (xy**) can make a low hand when at least three other low ranks beside x and y appear on the board. With two high cards in the hand, there are 52−4r−2=50−4r{\displaystyle 52-4r-2=50-4r} high cards and 4r−4+2=4r−2{\displaystyle 4r-4+2=4r-2} low cards left in the deck. The following tables show the derivations when holding two low ranks and two high cards.

Derivations for low hand shape xy** (two low ranks and two high cards) on the flop